\documentclass[../libro.tex]{subfiles}

\begin{document}

\ifSubfilesClassLoaded{\appendix\chapter{Afterstories}\clearpage}{}

\section{Until the re-seeing}

\begingroup
\setlength{\parindent}{1.5em}%for 12pt

% In this section, pi is upright.
\let\umathpi\pi
\renewcommand*{\pi}{\symup\umathpi}

% In this section, tau is 2pi, so it is also upright.
\let\umathtau\tau
\renewcommand*{\tau}{\symup\umathtau}

The title of this section is a literal translation of
``Ĝis la revido'' in Esperanto,
which idiomatically means ``See you'' in English.
Yes, I am bidding farewell---it is not the case that
\textit{everything} that has a beginning has an end,
but it \textit{is} the case that
this book,
\textit{An introduction to determinants},
which has a beginning, has an end.

I do not really have anything interesting to say,
but I still choose to say something here.

It is said that there are at least five ways
of defining the determinant:
\begin{itemize}
    \item as a function satisfying certain axioms;
    \item explicitly by the ``big formula'';
    \item recursively;
    \item as the ``oriented volume'' of a parallelepiped;
    \item by the exterior algebra.
\end{itemize}
I define the determinant recursively in this book.
I believe that this is the easiest way for beginners.
Some other ways are hinted in this book as well.

This book is mainly about determinants,
and it is over four hundred pages long.
I know that I have written too much in this book:
the number of pages in Appendix C
(the current chapter),
which consists of reading materials,
is greater than the total number of pages in all other chapters.
However, I really wanted to share my knowledge,
hoping that it might be useful and helpful.

This book is available under
the \href{https://opensource.org/license/0bsd}{Zero-Clause BSD License},
which means that
any person or organisation is allowed to use this book
in any way, for any purpose, with or without charge.
I have not yet earned a single cent from this book,
but I did not, do not, and will not
prevent any person or organisation
from making a profit from this book.
I want to propagate the values of ``open knowledge'',
and I want to prove its profitability,
if I can ever earn money from this book.

I thank mathematicians and their work;
I thank artists who made artworks for this book;
I thank my friends, who were good brokers;
I thank everyone who read or reads this book.

Parenthetically, the day on which
I wrote this closing remark is 28 June 2025.
Both six and twenty-eight are perfect numbers,
in the sense that
each is equal to the sum of all its positive divisors,
excluding itself.
It is also worth noting that
\(6.28\) is a decimal approximation of \(2\pi\)
(or \(\tau\)),
the ratio of the circumference of a circle to its radius.
Therefore I think of 28 June as
a ``mirinda'' (meaning ``worth marvelling at'') day.
Moreover, \(2\,025\) is remarkable:
\begin{align*}
    2\,025
    = {} &
    1^6 \cdot 3^4 \cdot 5^2 \cdot 7^0
    \\
    = {} &
    5^2 + 20^2 + 40^2
    \\
    = {} &
    27^2 + 36^2
    \\
    = {} &
    (20 + 25)^2
    \\
    = {} &
    (1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9)^2
    \\
    = {} &
    1^3 + 2^3 + 3^3 + 4^3 + 5^3 + 6^3 + 7^3 + 8^3 + 9^3
    \\
    = {} &
    \det \begin{bmatrix}
             1  & 5  & 14 & 12 \\
             9  & 2  & 7  & 6  \\
             13 & 8  & 3  & 11 \\
             15 & 10 & 4  & 16 \\
         \end{bmatrix}
    \\
    = {} &
    \det \begin{bmatrix}
             1  & 18 & 9  & 6  & 10 \\
             5  & 24 & 13 & 7  & 12 \\
             8  & 22 & 21 & 23 & 14 \\
             16 & 2  & 15 & 4  & 17 \\
             19 & 11 & 20 & 3  & 25 \\
         \end{bmatrix}.
\end{align*}
I have only listed a few here.

Fartu bone. Kredu, ke ni renkontiĝos iam, ie, iel, ial.
(Do well. Believe that we will meet
sometime, somewhere, somehow, for some reason.)

\endgroup

\end{document}

This is just a closing remark.

One can find a square matrix whose determinant = 2025
with the help of a computer.

Here is a Wolfram program.
One can run it with https://tio.run/#mathematica
if one does not have Wolfram locally on one's computer.
Note that this program does a randomised search.

\begin{lstlisting}

FindMatrixWithDeterminant[size_, target_, trials_:10^9] :=
    Module[
        {n = size^2, elements, mat, det, found = False}
        ,
        (*Generate the first size^2 positive integers, excluding 1 and size^2*)
        elements = Complement[Range[n], {1, n}];
        (*Perform randomised trials*)
        Do[(*Randomly permute the remaining elements and prepend 1 and append size^2*)
            mat = Partition[Join[{1}, RandomSample[elements], {n}], size];
            (*Compute the determinant*)
            det = Det[mat];
            (*Check if the determinant matches the target*)
            If[det == target,
                Print["A (an) ", size, " by ", size, " matrix whose det = ",
                     target, " (trial ", i, "):"];
                Print[MatrixForm[mat]];
                found = True;
                Return[]; (*Exit on success*)
            ]
            ,
            {i, trials}
        ];
        If[!found,
            Print["No such matrix has been found after ", trials, " trials."
                ]
        ];
    ]

FindMatrixWithDeterminant[4, 2025]

FindMatrixWithDeterminant[5, 2025]

\end{lstlisting}

% 2025
Det[{{1, 5, 14, 12}, {9, 2, 7, 6}, {13, 8, 3, 11}, {15, 10, 4, 16}}]

% 2025
Det[{{1, 18, 9, 6, 10}, {5, 24, 13, 7, 12}, {8, 22, 21, 23, 14}, {16, 2, 15, 4, 17}, {19, 11, 20, 3, 25}}]
